Global regularity and estimates for solutions of parabolic equations

V. I. Bogachev   ; M. Röckner   ; S. V. Shaposhnikov

Teoriâ veroâtnostej i ee primeneniâ, Tome 50 (2005) no. 4, pp. 652-674

Résumé

Given a second-order parabolic operator $$ Lu(t,x):=\frac{\partial u(t,x)}{\partial t}+a^{ij}(t,x)\partial_{x_i}\partial_{x_j}u(t,x)+b^i(t,x)\partial_{x_i}u(t,x), $$ we consider the weak parabolic equation $L^{*}\mu=0$ for Borel probability measures on $(0,1)\times\mathbf{R}^d$. The equation is understood as the equality $$ \int_{(0,1)\timesR^d} Lu\,d\mu=0 $$ for all smooth functions $u$ with compact support in $(0,1)\timesR^d$. This equation is satisfied for the transition probabilities of the diffusion process associated with $L$. We show that under broad assumptions, $\mu$ has the form $\mu=\varrho(t,x)\,dt\,dx$, where the function $x\mapsto\varrho(t,x)$ is Sobolev, $|\nabla_x \varrho(x,t)|^2/\varrho(t,x)$ is Lebesgue integrable over $[0,\tau]\times\mathbf{R}^d$, and $\varrho\in L^p([0,\tau]\timesR^d)$ for all $p\in[1,+\infty)$ and $\tau<1$. Moreover, a sufficient condition for the uniform boundedness of $\varrho$ on $[0,\tau]\timesR^d$ is given.

Détail

Keywords: parabolic equations for measures, transition probabilities, regularity of solutions of parabolic equations, estimates of solutions of parabolic equations.

Bibliographie

[1] Bogachev V. I., Da Prato G., Röckner M., “Existence of solutions to weak parabolic equations for measures”, Proc. London Math. Soc., 88:3 (2004), 753–774 | DOI | MR | Zbl

[2] Bogachev V. I., Da Prato G., Röckner M., On solvability of weak parabolic equations for measures, Preprint, Reseach Center BiBoS, Bielefeld University, Bielefeld, 2005

[3] Bogachev V. I., Krylov N. V., Röckner M., “On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions”, Comm. Partial Differential Equations, 26:11–12 (2001), 2037–2080 | DOI | MR | Zbl

[4] Stannat W., Time-dependent diffusion operators on $L^1$, Preprint SFB 343 No 00-080, Univ. Bielefeld, Bielefeld, 2000 | MR

[5] Bogachev V. I., Röckner M., “Regularity of invariant measures on finite- and infinite-dimensional spaces and applications”, J. Funct. Anal., 133:1 (1995), 168–223 | DOI | MR | Zbl

[6] Bogachev V. I., Krylov N. V., Röckner M., “Regularity of invariant measures: the case of non-constant diffusion part”, J. Funct. Anal., 138:1 (1996), 223–242 | DOI | MR | Zbl

[7] Bogachev V. I., Röckner M., Wang F.-Y., “Elliptic equations for invariant measures on finite and infinite dimensional manifolds”, J. Math. Pures Appl., 80:2 (2001), 177–221 | DOI | MR | Zbl

[8] Metafune G., Pallara D., Rhandi A., “Global regularity of invariant measures”, J. Funct. Anal., 223:2 (2005), 396–424 | DOI | MR | Zbl

[9] Bogachev V. I., Krylov N. V., Rekner M., “Regulyarnost i globalnye otsenki plotnostei invariantnykh mer diffuzionnykh protsessov”, DAN, 405:5 (2005), 583–587 | MR | Zbl

[10] Bogachev V. I., Krylov N. V., Röckner M., Elliptic equations for measures: regularity and global bounds of densities, Preprint No 05-06-186, Research Center BiBoS, Bielefeld University, Bielefeld; J. Math. Pures Appl. (9), 85:6 (2006), 743–757 | MR | Zbl

[11] Carlen E. A., “Conservative diffusions”, Comm. Math. Phys., 94:3 (1984), 293–315 | DOI | MR | Zbl

[12] Cattiaux P., Fradon M., “Entropy, reversible diffusion processes, and Markov uniqueness”, J. Funct. Anal., 138:1 (1996), 243–272 | DOI | MR | Zbl

[13] Cattiaux P., Léonard C., “Minimization of the Kullback information of diffusion processes”, Ann. Inst. H. Poincaré, 30:1 (1994), 83–132 ; correction: ibid. 31:4 (1995), 705–707 | MR | Zbl | MR | Zbl

[14] Föllmer H., “Random fields and diffusion processes”, Lecture Notes in Math., 1362, 1988, 101–203 | MR | Zbl

[15] Nelson E., Quantum Fluctuations, Princeton Univ. Press, Princeton, 1985, 147 pp. | MR | Zbl

[16] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967, 736 pp. | MR